The correct and unusual coordinate transformation rules for electromagnetic quadrupoles.

J Gratus,T Banaszek

doi:10.1098/rspa.2017.0652

Abstract

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.

Highlights

Multipole expansions are used extensively as an approximation of extended particles where the mass or charge is considered to be concentrated at one point.2018 The Authors
Even electric dipoles and electric quadrupoles can be defined without reference to a metric
We show which quadrupoles are dipoles and which quadrupoles are electric dipoles

Summary

Introduction

Multipole expansions are used extensively as an approximation of extended particles where the mass or charge is considered to be concentrated at one point. Since the correct coordinate transformations for quadrupoles have been unknown up to now, the use of multipole expansions has been limited to Cartesian coordinates in flat space. The quadrupole which flows unchanged in the adapted coordinates gains a dipole in the laboratory frame Such a fluid flow is uncommon in electromagnetism, it is the natural Vlasov description for the dynamics of a distribution of charge in seven-dimensional phase–space–time. He derived the electromagnetic fields due to an arbitrary moving multipole in Minkowski space–time by differentiating the Liénard–Wiechart fields As stated, these results require that the quadrupole is expressed in Cartesian coordinates. The magnetic multipoles require either a metric or a preferred coordinate system to define them The advantage of such definitions are many fold:. In appendix A, we prove some of the more technical statements from §4

The standard representation of quadrupoles

Coordinate transformations of quadrupoles

Coordinate-free and metric-free definition of multipoles

Conclusion and discussion